Solve -2a/4+3a^2/-2a=1

$$\frac { - 2 a } { 4 } + \frac { 3 a ^ { 2 } } { - 2 a } = 1$$

Solve for a

$$a=-\frac{1}{2}=-0.5$$

Steps Using Factoring

Variable a cannot be equal to 0 since division by zero is not defined. Multiply both sides of the equation by 4a, the least common multiple of 4,-2a.

$$a\left(-2\right)a-2\times 3a^{2}=4a$$

Multiply a and a to get a^{2}.

$$a^{2}\left(-2\right)-2\times 3a^{2}=4a$$

Multiply -2 and 3 to get -6.

$$a^{2}\left(-2\right)-6a^{2}=4a$$

Combine a^{2}\left(-2\right) and -6a^{2} to get -8a^{2}.

$$-8a^{2}=4a$$

Subtract 4a from both sides.

$$-8a^{2}-4a=0$$

Factor out a.

$$a\left(-8a-4\right)=0$$

To find equation solutions, solve a=0 and -8a-4=0.

$$a=0 a=-\frac{1}{2}$$

Variable a cannot be equal to 0.

$$a=-\frac{1}{2}$$

Steps Using the Quadratic Formula

Variable a cannot be equal to 0 since division by zero is not defined. Multiply both sides of the equation by 4a, the least common multiple of 4,-2a.

$$a\left(-2\right)a-2\times 3a^{2}=4a$$

Multiply a and a to get a^{2}.

$$a^{2}\left(-2\right)-2\times 3a^{2}=4a$$

Multiply -2 and 3 to get -6.

$$a^{2}\left(-2\right)-6a^{2}=4a$$

Combine a^{2}\left(-2\right) and -6a^{2} to get -8a^{2}.

$$-8a^{2}=4a$$

Subtract 4a from both sides.

$$-8a^{2}-4a=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute -8 for a, -4 for b, and 0 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$a=\frac{-\left(-4\right)±\sqrt{\left(-4\right)^{2}}}{2\left(-8\right)}$$

Take the square root of \left(-4\right)^{2}.

$$a=\frac{-\left(-4\right)±4}{2\left(-8\right)}$$

The opposite of -4 is 4.

$$a=\frac{4±4}{2\left(-8\right)}$$

Multiply 2 times -8.

$$a=\frac{4±4}{-16}$$

Now solve the equation a=\frac{4±4}{-16} when ± is plus. Add 4 to 4.

$$a=\frac{8}{-16}$$

Reduce the fraction \frac{8}{-16} to lowest terms by extracting and canceling out 8.

$$a=-\frac{1}{2}$$

Now solve the equation a=\frac{4±4}{-16} when ± is minus. Subtract 4 from 4.

$$a=\frac{0}{-16}$$

Divide 0 by -16.

$$a=0$$

The equation is now solved.

$$a=-\frac{1}{2} a=0$$

Variable a cannot be equal to 0.

$$a=-\frac{1}{2}$$

Steps for Completing the Square

Variable a cannot be equal to 0 since division by zero is not defined. Multiply both sides of the equation by 4a, the least common multiple of 4,-2a.

$$a\left(-2\right)a-2\times 3a^{2}=4a$$

Multiply a and a to get a^{2}.

$$a^{2}\left(-2\right)-2\times 3a^{2}=4a$$

Multiply -2 and 3 to get -6.

$$a^{2}\left(-2\right)-6a^{2}=4a$$

Combine a^{2}\left(-2\right) and -6a^{2} to get -8a^{2}.

$$-8a^{2}=4a$$

Subtract 4a from both sides.

$$-8a^{2}-4a=0$$

Divide both sides by -8.

$$\frac{-8a^{2}-4a}{-8}=\frac{0}{-8}$$

Dividing by -8 undoes the multiplication by -8.

$$a^{2}+\left(-\frac{4}{-8}\right)a=\frac{0}{-8}$$

Reduce the fraction \frac{-4}{-8} to lowest terms by extracting and canceling out 4.

$$a^{2}+\frac{1}{2}a=\frac{0}{-8}$$

Divide 0 by -8.

$$a^{2}+\frac{1}{2}a=0$$

Divide \frac{1}{2}, the coefficient of the x term, by 2 to get \frac{1}{4}. Then add the square of \frac{1}{4} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$a^{2}+\frac{1}{2}a+\left(\frac{1}{4}\right)^{2}=\left(\frac{1}{4}\right)^{2}$$

Square \frac{1}{4} by squaring both the numerator and the denominator of the fraction.

$$a^{2}+\frac{1}{2}a+\frac{1}{16}=\frac{1}{16}$$

Factor a^{2}+\frac{1}{2}a+\frac{1}{16}. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(a+\frac{1}{4}\right)^{2}=\frac{1}{16}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(a+\frac{1}{4}\right)^{2}}=\sqrt{\frac{1}{16}}$$

Simplify.

$$a+\frac{1}{4}=\frac{1}{4} a+\frac{1}{4}=-\frac{1}{4}$$

Subtract \frac{1}{4} from both sides of the equation.

$$a=0 a=-\frac{1}{2}$$

Variable a cannot be equal to 0.

$$a=-\frac{1}{2}$$